
\(\Lambda \) CDM model with a equations of state (EoS) of the vacuum energy \(w=-1\)  is the most promising candidate of dark energy. Its normalized Hubble parameter defined as \(E(z)\equiv H(z)/H_0\)  is
\(E(z)=\sqrt{\Omega _{\mathrm {m}}(1+z)^{3}+\Omega _{\mathrm {r}}(1+z)^{4}+\left(1-\Omega _{\mathrm {m}}-\Omega _{\mathrm {r}}\right)}.\) 

\(w\) CDM model with a constant EoS \(w\)  is the simplest case for dynamical dark energy and its normalized Hubble parameter is given by
\(E(z)= \sqrt{\Omega _{m}(1+z)^{3}+\Omega _{r}(1+z)^{4}+\left(1-\Omega _{m}-\Omega _{r}\right)(1+z)^{3(1+w)}}.\) 

CPL model is a phenomenological model to explore the evolution of \(w\) . For this model, the forms of \(w(z)\)  and normalized Hubble parameter are written as, respectively,
\(w(z)=w_0+w_a\frac{z}{1+z},\) 
\(\begin{aligned}E(z)&=&\sqrt{ \Omega _{m}(1+z)^{3}+\Omega _{r}(1+z)^{4}+\left(1-\Omega _{m}-\Omega _{r}\right)(1+z)^{3\left(1+w_{0}+w_{a}\right)} \exp \left(-\frac{3 w_{a} z}{1+z}\right)}.\end{aligned}\) 

The holographic dark energy model is constructed by considering the holographic principle of quantum gravity theory in a quantum effective field theory [1]}. HDE model could effectively alleviate the fine-tuning and coincidence problems that are serious problems in the \(\Lambda \) CDM model [1]}. In this model, the density of dark energy is given by
\(\rho _{\mathrm {de}} \propto M_{\mathrm {pl}}^{2} L^{-2},\) 
where \(M_{\mathrm {pl}}\)  is the reduced Planck mass, and \(L\)  is the infrared (IR) cutoff length scale in the effective quantum field theory. Different holographic dark energy models depend on the choice of different IR cutoff \(L\)  [3]}, [4]}, [5]}, [6]}, [7]}. In this paper, we choose the event horizon size of the universe as the IR cutoff [1]}. In this case, the dark energy density is given by
\(\rho _{\mathrm {de}}=3 c^2 M_{\mathrm {pl}}^{2} R_{\mathrm {eh}}^{-2},\) 
where \(c\)  is a dimensionless phenomenological parameter determining properties of the HDE model, and it should not be confused with the speed of light.
\(R_{\mathrm {eh}}\)  is the event horizon size defined as
\(R_{\mathrm {eh}}(t)=a(t) \int _{t}^{\infty } \frac{d t^{\prime }}{a\left(t^{\prime }\right)}.\) 
The evolution of the HDE obeys the following differential equations
\(\begin{aligned}&\frac{1}{E(z)} \frac{d E(z)}{d z}=-\frac{\Omega _{\mathrm {de}}(z)}{1+z}\left(\frac{1}{2}+\frac{\sqrt{\Omega _{\mathrm {de}}(z)}}{c}-\frac{3}{2 \Omega _{\mathrm {de}}(z)}\right), \\&\frac{d \Omega _{\mathrm {de}}(z)}{d z}=-\frac{2 \Omega _{\mathrm {de}}(z)\left(1-\Omega _{\mathrm {de}}(z)\right)}{1+z}\left(\frac{1}{2}+\frac{\sqrt{\Omega _{\mathrm {de}}(z)}}{c}\right).\end{aligned}\) 

Assuming some coupling between dark energy and cold dark matter, the interacting dark energy models were proposed, in which the energy conservation equations for dark energy and cold dark matter usually satisfy
\(\dot{\rho }_{\mathrm {de}}&=&-3 H(1+w) \rho _{\mathrm {de}}+Q, \\\dot{\rho }_{\mathrm {c}}&=&-3 H \rho _{\mathrm {c}}-Q, \) 
where \({\rho }_{\mathrm {de}}\)  and \({\rho }_{\mathrm {c}}\)  are the energy densities of dark energy and cold dark matter, respectively, and a dot represents the derivative with respect to the cosmic time. For the energy transfer rate \(Q\) , its specific form is still an open question and many forms have been discussed in previous works [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}. In this paper, we consider a phenomenological form of \(Q=\beta H\rho _{\rm {de}}\) , where \(\beta \)  is a dimensionless coupling parameter. Thus, once the specific form of \(Q\)  is determined, the expression of \(E^2(z)\)  can be obtained by solving the differential equations (REF ) and (),
\(\begin{aligned}E^{2}(z)=& \Omega _{\operatorname{de}}\left(\frac{\beta }{w+\beta }(1+z)^{3}+\frac{w}{w+\beta }(1+z)^{3(1+w+\beta )}\right) \\&+\Omega _{m}(1+z)^{3}+\Omega _{r}(1+z)^{4},\end{aligned}\) 
where \(\Omega _{m}=\Omega _{c}+\Omega _{b}\) . Here, we just consider the case of I\(\Lambda \) CDM model corresponding to \(w=-1\) .

